|
|
A245055
|
|
Decimal expansion of 'tau' (named sigma_2 by C. Pomerance), a constant associated with the expected number of random elements to generate a finite abelian group.
|
|
0
|
|
|
1, 7, 4, 2, 6, 5, 2, 3, 1, 1, 0, 3, 3, 5, 1, 5, 4, 3, 5, 2, 4, 8, 9, 0, 4, 8, 0, 6, 9, 4, 1, 2, 9, 8, 6, 4, 1, 1, 5, 4, 4, 3, 7, 9, 8, 9, 8, 3, 8, 1, 0, 4, 6, 2, 8, 1, 4, 2, 9, 0, 4, 7, 9, 5, 7, 4, 6, 5, 5, 5, 0, 3, 8, 7, 0, 0, 8, 1, 3, 5, 0, 8, 6, 8, 0, 5, 8, 1, 4, 7, 4, 1, 7, 5, 2, 4, 7, 8, 8, 1, 2
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
REFERENCES
|
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.1 Abelian group enumeration constants, p. 273.
|
|
LINKS
|
|
|
FORMULA
|
tau = sum_{j >= 1} (1-(1-2^(-j))*prod_{k >= j+1} zeta(k)^(-1)).
tau = sum_{j >= 1} (1-(1-2^(-j))*c*prod_{k = 2..j} zeta(k)), where c is A068982.
|
|
EXAMPLE
|
1.7426523110335154352489048069412986411544379898381...
|
|
MATHEMATICA
|
digits = 101; max = 400; c = 1/Product[N[Zeta[k], digits + 100], {k, 2, max}]; p[j_] := Product[N[Zeta[k], digits + 100], {k, 2, j}]; tau = Sum[1 - (1 - 2^-j)*c*p[j], {j, 1, max}]; RealDigits[tau, 10, digits ] // First
|
|
PROG
|
(PARI) default(realprecision, 120); suminf(j=1, 1-(1-2^(-j))*prodinf(k=j+1, 1/zeta(k))) \\ Vaclav Kotesovec, Oct 22 2014
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|